Date: May 18, 2013 7:39 PM
Author: fom
Subject: Re: A logically motivated theory
On 5/18/2013 2:43 PM, Zuhair wrote:> On May 18, 8:58 pm, fom <fomJ...@nyms.net> wrote:>> On 5/18/2013 10:40 AM, Zuhair wrote:>>>>>>> The whole motivation beyond this theory is to extend any first order>>> predicate by objects.>>>> Could you please clarify this remark?>> I'll expand on that. The motivation of this theory is to 'define'> objects that uniquely corresponds to first order logic predicates,i.e> for each first order predicates P,Q there exist objects P* and Q* such> that P*=Q* iff for all x. P(x)<->Q(x)>> We want to do that for EVERY first order predicate P.>> The plan is to stipulate the existence of multiple PRIMITIVE binary> extensional relations, each one of those would play the role of a set> membership relation after which object representative of predicates> are defined.>> Now the first axiom ensures that no 'distinct' objects can be defined> after equivalent predicates all across the membership relations, so> although we have many membership relation but yet any objects X,Y that> are co-extensional over relations E,D (i.e. for all z. z E X iff z D> Y) are identical!> The second axiom (schema of course) ensures that no object represent> non-equivalent Predicates, and so although we do have 'multiple'> membership relations (primitive extensional relations) however from> axiom schemas 1 and 2 this would ensure that each object defined after> any of those relations would stand 'uniquely' for a single predicate.>> The last axiom scheme is just a statement ensuring the existence of an> object that extends each first order predicate after some membership> relation.>> Those objects uniquely corresponding for first order predicates are to> be called as: Sets.>> The point is that paradoxes are eliminated because of having> 'multiple' extensional relations each standing as a membership> relation.>> I'm not sure if that would interpret PA, but if it does, then PA can> be said to be a PURELY logical theory!>> Now if (and this is a big if) we allow infinitely long formulas to> define first order predicates (infinitary first order languages) then> second order arithmetic 'might' follow. And I think if this is the> case, then second order arithmetic is also PURELY logical!>> This mean that the bulk of traditional mathematics (most of which can> be formulated within proper subsets of second order arithmetic) is> purely logical!>> However I don't think that higher mathematics can have a pure logical> motivation comparable to the above, the motivation behind those can be> said to be 'structural', or 'constructive' in the ideal sense that> I've presented in my latest philosophical notes on my website and to> this Usenet.>> Zuhair>